Left invertible quasi-isometric liftings

TL;DR

This paper investigates the existence and properties of left invertible quasi-isometric liftings for certain operators in Hilbert spaces, extending the theory of isometric liftings to broader classes like quasicontractions.

Contribution

It establishes the existence of such liftings for operators similar to contractions and characterizes those admitting minimal quasi-isometric liftings within the framework of A-contractions.

Findings

Existence of left invertible quasi-isometric liftings for specific operator classes

Characterization of operators with minimal quasi-isometric liftings

Analysis of matrix structures of liftings and their parallels with isometric liftings

Abstract

Quasi-isometric liftings similar to isometries, for the operators similar to contractions in Hilbert spaces, are investigated. The existence of such liftings is established, and their applications are explored for specific operator classes, including quasicontractions. A particular focus is placed on operators that admit left invertible minimal quasi-isometric liftings. These operators are characterized within the framework of $A$-contractions, and the matrix structures of their liftings are analyzed, highlighting parallels with the isometric liftings of contractions.